Characterizing rings in terms of the extent of the injectivity and projectivity of their modules. (English) Zbl 1284.16003
For a ring \(R\), the module \(M\) is injective relative to \(N\) if for any submodule \(K\) of \(N\) the induced map \(\operatorname{Hom}_R(N,M)\to\operatorname{Hom}_R(K,M)\) is surjective.
For fixed \(M\), the injectivity domain \(\mathcal In^{-1}(M)\) is the collection of all \(N\), such that \(M\) is relative injective to \(N\). One may note that the injectivity domain of \(M\) is the entire module category if and only if \(M\) is injective.
Finally, the i-profile of \(R\) is the collection of all injectivity domains, that is \(\{\mathcal In^{-1}(M)\mid M\in\text{Mod-}R\}\).
The paper under review studies the structure of i-profiles of rings (as posets), and of p-profiles (defined dually via relative projective modules). At the same time the authors also consider the question what properties of the poset (for instance it being linearly ordered) mean for the structure of the ring \(R\).
For fixed \(M\), the injectivity domain \(\mathcal In^{-1}(M)\) is the collection of all \(N\), such that \(M\) is relative injective to \(N\). One may note that the injectivity domain of \(M\) is the entire module category if and only if \(M\) is injective.
Finally, the i-profile of \(R\) is the collection of all injectivity domains, that is \(\{\mathcal In^{-1}(M)\mid M\in\text{Mod-}R\}\).
The paper under review studies the structure of i-profiles of rings (as posets), and of p-profiles (defined dually via relative projective modules). At the same time the authors also consider the question what properties of the poset (for instance it being linearly ordered) mean for the structure of the ring \(R\).
Reviewer: Steffen Oppermann (Trondheim)
MSC:
| 16D50 | Injective modules, self-injective associative rings |
| 16D40 | Free, projective, and flat modules and ideals in associative algebras |
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